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Henri Poincaré, Why space with three dimensions – Space and senses

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The question seems resolved; we seem to have only to apply this rule, either to the physical continuum, which is the coarse image of space, or to the corresponding mathematical continuum which is its refined image and which is the geometer’s space. This is an illusion; it would be fine if the physical continuum from which we draw space was directly given to us by the senses, but it is far from being so.

Let us see, indeed, how one can, from the mass of our sensations, deduce a physical continuum. Each element of a physical continuum is a set of sensations; and the simplest is to first consider a set of simultaneous sensations, a state of consciousness. But each of our states of consciousness is something excessively complex, so that we can never hope to see two states of consciousness become indistinguishable, and yet to construct a physical continuum, it is essential, from what precedes, that two of its elements may, in certain cases, be regarded as indistinguishable. Now, it will never happen that we can say: I can not discern my current state of mind from my state of mind the day before yesterday at such an hour.

It is therefore necessary that, by an active operation of the mind, we agree to consider two states of consciousness as identical by disregarding their differences. We can, for example, and it is the simplest way, to disregard the data of certain senses. I said that I could not distinguish a weight of 10 grams from a weight of 11 grams; it is probable, however, that if I have ever had the experience, the sensation of pressure caused by the weight of 10 grams was accompanied by various olfactory or auditory sensations, and that when the weight of 10 grams was replaced by that of 11 grams. these different sensations had varied; it is because I disregarded from these foreign sensations, that I could say that the two states of consciousness were indiscernible.

Other more complicated conventions can be made; we can also consider as elements of our continuum, not only sets of simultaneous sensations, but sets of successive sensations, sequences of sensations. It will then be necessary to make the fundamental convention and to say which are the common characters that must have two elements of the continuum (that they are two sets of simultaneous or successive sensations), so that one must consider them as identical.

Thus, for the definition of a physical continuum, it is necessary to make a double choice: 1 ° to choose the sets of simultaneous or successive sensations that must serve as elements of this continuum; 2 ° to choose the fundamental convention which will define the cases where two elements must be regarded as identical.

How to make this double choice to get the space? Can we be content to consider a set of simultaneous sensations or should we consider a series of sensations? Can we, in particular, content ourselves with the simplest, most natural fundamental convention of abstracting the data of certain senses? No.

Such an abstraction is impossible, we can not choose among our senses those who will give us all the space and give us only that; there is not one who can give us space without the help of others; nor is it one that gives us a lot of things that have nothing to do with space.

If we analyze, for example, the data of the proper touch, here is what we see; experience shows us that if we touch the skin with two points, the consciousness distinguishes these two points if they are sufficiently distant from each other and ceases to distinguish them if they are very close together; the minimum distance which makes it possible to discern them varies according to the regions of the body; it is usually said that the skin is divided into departments, each of which is the domain of the same sensory nerve; that if the two points fall into the same area, a single nerve is shaken, and we perceive only one point; but that we perceive two on the contrary if they fall into two areas and affect, consequently, two nerves. This is not entirely satisfactory; we would not thus find the characters of the physical continuum; suppose that we move the two points, their distance, besides very small, being kept constant. This distance being very small, we will have chances for them to fall in the same department and to have only one perception; but if we move them little by little without changing their distance, there must come a time when one of them will be out of the area and the other will not yet be out of the area. At that moment one should feel two points; but that is not what we observe; we would not obtain the notion of a physical continuum, but that of a discrete set formed of as many distinct individuals as there are areas. It is better to admit that the contact of a point affects, not only the nearest nerve, but also the neighboring nerves, and that with an intensity which decreases when the distance increases. Suppose, then, that the effects of the contact of two points are compared; if the distance between the two points is small, the same nerves are affected; the intensity of the excitation of one and the same nerve by the one and the other will doubtless be different, but this difference will be too feeble to be discerned, according to the general rule of Fechner. If a nerve is affected by the point A, but not by the point B, it will be very little by the point A and the excitation will be below the “threshold of consciousness”. The effects of the two points will therefore be indistinguishable.

We have then all that is necessary to build a physical continuum, we have only to walk two points on the surface of the skin and to note the cases where our conscience distinguishes them. We have abstracted (and this is what I referred to above as our fundamental convention) from a lot of circumstances, from the intensity of the shaking of each sensory thread; the more or less great pressure exerted on the skin by the point, of the nature of the contact; all these circumstances are revealed to us by touch, but we have eliminated them to preserve only those whose character is geometric. Do we have space? No ; firstly, the continuum thus constructed has only two dimensions, like the surface of the skin itself; then we know that our skin is mobile, that the same point of the skin does not always correspond to the same point of space; that the distance of two points of our skin varies when our body is deformed. This is probably how molluscs design space, but that has nothing to do with ours.

For sight, it’s the same thing; two beams of light striking two points of the retina, will give us the impression of two spots of light or of one, according as these two points will be more or less distant. We have the equivalent of our two points just now; we can use it to construct a physical continuum by disregarding the color and intensity of light; this physical continuum will have two dimensions like the surface of the retina. We will introduce the third dimension by involving the convergence of the eyes in the binocular vision, and this is what we called the visual space. It is superior to the tactile space, first of all because with a little good will, we can give it three dimensions, and secondly because the retina is probably mobile, but like a solid body, while the skin can bend in all directions. We are then tempted to say that this is the real space where we seek to locate all our other sensations. It’s not going well yet; not only is the eye mobile, so that at the same point of the retina, at the same degree of convergence of the eyes, does not always correspond to the same point of space; but we do not explain why we have introduced a third dimension, so obviously heterogeneous to the other two, nor why the geometry of the blind is the same as ours.

If we combine visual space with tactile space, we will have 5 dimensions, instead of 3 or 2; and it will remain to explain by what process these 5 dimensions are reduced to 3; and the number of dimensions will be increased if we want to bring other senses into the combination.

It remains to explain in a word why the tactile space and the visual space are one and the same space.

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